Lemma 6.24.2. Let $f : X \to Y$ be a continuous map of topological spaces. Let $\mathcal{O}$ be a presheaf of rings on $Y$. Let $\mathcal{G}$ be a presheaf of $\mathcal{O}$-modules. There is a natural map of underlying presheaves of sets

\[ f_ p\mathcal{O} \times f_ p\mathcal{G} \longrightarrow f_ p\mathcal{G} \]

which turns $f_ p\mathcal{G}$ into a presheaf of $f_ p\mathcal{O}$-modules. This construction is functorial in $\mathcal{G}$.

**Proof.**
Let $U \subset X$ is open. We define the map of the lemma to be the map

\begin{eqnarray*} f_ p\mathcal{O}(U) \times f_ p\mathcal{G}(U) & = & \mathop{\mathrm{colim}}\nolimits _{f(U) \subset V} \mathcal{O}(V) \times \mathop{\mathrm{colim}}\nolimits _{f(U) \subset V} \mathcal{G}(V) \\ & = & \mathop{\mathrm{colim}}\nolimits _{f(U) \subset V} (\mathcal{O}(V)\times \mathcal{G}(V)) \\ & \to & \mathop{\mathrm{colim}}\nolimits _{f(U) \subset V} \mathcal{G}(V) \\ & = & f_ p\mathcal{G}(U). \end{eqnarray*}

Here the arrow in the middle is the multiplication map on $Y$. The second equality holds because directed colimits commute with finite limits, see Categories, Lemma 4.19.2. We leave it to the reader to see this is compatible with restriction mappings and defines a structure of $f_ p\mathcal{O}$-module on $f_ p\mathcal{G}$.
$\square$

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